structures:matrix:linear
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| Within a **[[#space]]**, given a **[[#basis]]**, each **[[linear#linear map|linear transformation]]** can be represented by a square matrix and every appropriately sized matrix is a linear transformation. Transformations can be thought of as mappings, or as **[[#functions]]**. Matrices can represent these very usefully. In two dimensional space here are some examples: | Within a **[[#space]]**, given a **[[#basis]]**, each **[[linear#linear map|linear transformation]]** can be represented by a square matrix and every appropriately sized matrix is a linear transformation. Transformations can be thought of as mappings, or as **[[#functions]]**. Matrices can represent these very usefully. In two dimensional space here are some examples: | ||
| - | \(\left[\matrix{0 & 0 \cr 0 & 1 }\right],\quad\) Projection onto vertical axis\\ \\ | + | \(\left[\matrix{3 & 0 \cr 0 & 3 }\right],\quad\) Scale by 3 around origin\\ \\ |
| - | \(\left[\matrix{1 & 2 \cr 0 & 1 }\right],\quad\) Horizontal shear\\ \\ | + | \(\left[\matrix{0 & -1 \cr 1 & 0 }\right],\quad\) Rotate right angle anticlockwise around origin\\ \\ |
| - | \(\left[\matrix{3 & 0 \cr 0 & \frac12 }\right],\quad\) Squeeze (asymetrically)\\ \\ | + | \(\left[\matrix{1 & 0 \cr 0 & -1 }\right],\quad\) Reflect on horizontal axis (negate vertical axis)\\ \\ |
| - | \(\left[\matrix{\cos2\theta & \sin2\theta \cr \sin2\theta & -\cos2\theta}\right],\quad\) Reflection across line through origin, angle `theta` to horizontal | + | \(\left[\matrix{-1 & 0 \cr 0 & -1 }\right],\quad\) Rotate half circle around origin (reflect vert then horiz)\\ \\ |
| - | + | \(\left[\matrix{0 & 0 \cr 0 & 1 }\right],\quad\) Projection onto vertical axis (zero the horiz component)\\ \\ | |
| - | ===linear transformations with matrices=== | + | \(\left[\matrix{1 & \frac12 \cr 0 & 1 }\right],\quad\) Horizontal shear (add half vert component to horiz one)\\ \\ |
| - | Now consider [[applications#square matrices]]. For matrix multiplication with `n`-dimenional square matrices we have an //Identity//, the //Unit// for multiplication, like `1` in the scalars, | + | |
| - | \[%%I=\left[\matrix{ | + | |
| - | 1&0&\cdots&0\cr | + | |
| - | 0&1&\ddots&\vdots\cr | + | |
| - | \vdots&\ddots&\ddots&0\cr | + | |
| - | 0&\cdots&0&1\cr | + | |
| - | }\right]\ \text{ because }\ IA=AI=A=1A.%%\] | + | |
| - | Consider: ` X=xI=[ [x,0],[0,x] ],` 2\({\times}\)2 matrix `A` and the 2D vector where `V` is \(\bf v\) written as single column matrix. \(\ xA=XA=AX\ \) and \(x\bf v\) corresponds to \(XV\). | + | |
| - | + | ||
| - | `X` is the simple linear transformation in 2D geometric space of scaling by `x` around the origin. In 2D vector space it is stretching the vectors by `x`, identical to scalar multiplication. Multiplying another 2\({\times}\)2 matrix results in composing the transformations into a single transformation, in the case of `X` it is the same as scalar multiplication by `x`. | + | |
| - | + | ||
| - | Within a **[[#space]]**, given a **[[#basis]]**, each **[[linear#linear map|linear transformation]]** can be represented by a square matrix and every appropriately sized matrix is a linear transformation. Transformations can be thought of as mappings, or as **[[#functions]]**. Matrices can represent these very usefully. In two dimensional space here are some examples: | + | |
| - | + | ||
| - | \(\left[\matrix{0 & 0 \cr 0 & 1 }\right],\quad\) Projection onto vertical axis\\ \\ | + | |
| - | \(\left[\matrix{1 & 2 \cr 0 & 1 }\right],\quad\) Horizontal shear\\ \\ | + | |
| \(\left[\matrix{3 & 0 \cr 0 & \frac12 }\right],\quad\) Squeeze (asymetrically)\\ \\ | \(\left[\matrix{3 & 0 \cr 0 & \frac12 }\right],\quad\) Squeeze (asymetrically)\\ \\ | ||
| \(\left[\matrix{\cos2\theta & \sin2\theta \cr \sin2\theta & -\cos2\theta}\right],\quad\) Reflection across line through origin, angle `theta` to horizontal | \(\left[\matrix{\cos2\theta & \sin2\theta \cr \sin2\theta & -\cos2\theta}\right],\quad\) Reflection across line through origin, angle `theta` to horizontal | ||
| Continued [[discussion#complex-as-matrix|in the main discussion]]. | Continued [[discussion#complex-as-matrix|in the main discussion]]. | ||
structures/matrix/linear.1740899592.txt.gz · Last modified: 2025/03/02 18:13 by simon
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